REFINEMENTS, EXTENSIONS AND GENERALIZATIONS OF THE SECOND KERSHAW’S DOUBLE INEQUALITY FENG QI Abstract. In the paper, the second Kershaw’s double inequality concerning ratio of two gamma functions is refined, extended and generalized elegantly.
1. Introduction In [13], the following two double inequalities were established: !1−s à r µ ¶1−s s 1 Γ(x + 1) 1 x+ < < x− + s+ , 2 Γ(x + s) 2 4 · µ ¶¸ £ ¡ √ ¢¤ Γ(x + 1) s+1 exp (1 − s)ψ x + s < < exp (1 − s)ψ x + , Γ(x + s) 2
(1) (2)
where 0 < s < 1, x ≥ 1, Γ is the classical Euler’s gamma function, and ψ is the logarithmic derivative of Γ. They are called the first and second Kershaw’s double inequality respectively. In [2, Theorem 2.7], the double inequality (2) was generalized to ¯ ¯ ¯ ¯ ¯ (n+1) ¯ ¯ψ (n) (x)¯ − ¯ψ (n) (y)¯ ¯ ¯ ¯ ¯ −ψ (L−(n+2) (x, y)) < < −¯ψ (n+1) (A(x, y))¯, (3) x−y where x and y are positive numbers, n is a positive integer, and Lp (a, b) defined in [4, p. 385] by · ¸1/p bp+1 − ap+1 , p 6= −1, 0 (p + 1)(b − a) b−a , p = −1 Lp (a, b) = (4) ln b − ln a µ ¶1/(b−a) 1 bb , p=0 e aa stands for the generalized logarithmic mean of √ order p ∈ R for positive numbers a and b with a 6= b. Note that L−2 (a, b) = ab = G(a, b), L−1 (a, b) = L(a, b), L0 (a, b) = I(a, b) and L1 (a, b) = a+b 2 = A(a, b) are called respectively the geometric mean, the logarithmic mean, the identric or exponential mean and the arithmetic Date: This manuscript was completed on 28 January 2007. 2000 Mathematics Subject Classification. 26A48, 26A51, 26D20, 33B10, 33B15, 65R10. Key words and phrases. Kershaw’s double inequality, refinement, extension, generalization, generalized logarithmic mean, ratio of two gamma functions, psi function, polygamma function, inequality, monotonicity. This paper was typeset using AMS-LATEX. 1
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mean. It is known [4, pp. 386–387, Theorem 3] that the generalized logarithmic mean Lp (a, b) of order p is increasing in p for a 6= b. Therefore, inequalities G(a, b) < L(a, b) < I(a, b) < A(a, b)
(5)
are valid for a > 0 and b > 0 with a 6= b. See also [26, 27]. There have been a lot of literature about these two double inequalities and their history, background, refinements, extensions, generalizations and applications. For more information, please refer to [5, 7, 9, 10, 11, 12, 14, 19, 20, 21, 22, 23, 30, 32, 34, 35, 36, 38, 39, 40, 41, 49] and the references therein. In [23], the following generalization, extension and refinement of the second Kershaw’s double inequality (2) were obtained: For s, t ∈ R with s 6= t, the function ·
¸1/(s−t) Γ(x + s) 1 (6) Γ(x + t) eψ(L(s,t;x)) is decreasing in x > − min{s, t}. In particular, for s, t ∈ R and x > − min{s, t} with s 6= t, · ¸1/(s−t) Γ(x + s) ψ(L(s,t;x)) e < < eψ(A(s,t;x)) , (7) Γ(x + t) where L(s, t; x) = L(x + s, x + t) and A(s, t; x) = A(x + s, x + t) for s, t ∈ R and x > − min{s, t} with s 6= t. In [41], the right hand side inequalities in (3) and (7) were refined as follows: For s, t ∈ R with s 6= t and x > − min{s, t}, inequalities · ¸1/(s−t) Γ(x + s) ≤ eψ(I(s,t;x)) (8) Γ(x + t) and
£ ¤ (−1)n ψ (n−1) (x + s) − ψ (n−1) (x + t) ≤ (−1)n ψ (n) (I(s, t; x)), (9) s−t hold, where I(s, t; x) = I(x + s, x + t) and n ∈ N. These inequalities (8) and (9) refine, extend and generalize the right hand side inequality in the second Kershaw’s double inequality (2). The aim of this paper is to generalize, extend and refine the right hand side inequalities in (2), (3) and (7). Meanwhile, the left hand side inequalities in (3) and (7) and inequalities (8) and (9) are recovered. The main results of this paper are the following theorems. Theorem 1. For real numbers s > 0 and t > 0 with s 6= t and an integer i ≥ 0, inequality Z (−1)i t (i) i (i) ψ (u) du ≤ (−1)i ψ (i) (Lq (s, t)) (10) (−1) ψ (Lp (s, t)) ≤ t−s s holds if p ≤ −i − 1 and q ≥ −i. Theorem 2. For s, t ∈ R with s 6= t and x > − min{s, t}, inequality ¸1/(s−t) · Γ(x + s) ψ(Lp (s,t;x)) < eψ(Lq (s,t;x)) e < Γ(x + t) holds if p ≤ −1 and q ≥ 0.
(11)
REFINEMENTS AND GENERALIZATIONS OF KERSHAW’S DOUBLE INEQUALITY
Theorem 3. For s, t ∈ R with s 6= t and x > − min{s, t}, the function · ¸ Z t 1 ψ (i) (x + u) du (−1)i ψ (i) (Lp (s, t; x)) − t−s s
3
(12)
is increasing in x if either p ≤ −(i + 2) or p = −(i + 1) and decreasing in x if p ≥ 1, where i ≥ 0 is an integer. Remark 1. It is conjectured that the function (12) is decreasing (even completely monotonic) in x if p ≥ −i and that its negative is decreasing (even completely monotonic) in x if p ≤ −(i + 1). 2. Proofs of theorems Proof of Theorem 1. It is apparent that the function f (x) = (−1)i ψ (i) (x) for i ≥ 0 is strictly increasing, the function g(x) = xp for p 6= 0 is monotonic in (0, ∞), and the inverse function of g is g −1 (x) = x1/p . Straightforward computation gives µ ¶ Z t 1 −1 g g(u) du = Lp (s, t), (13) t−s s ¡ ¢ h(x) , f ◦ g −1 (x) = (−1)i ψ (i) x1/p (14) and ¡ ¢¤ x1/p−2 £ 1/p (i+2) ¡ 1/p ¢ x ψ x − (p − 1)ψ (i+1) x1/p p2 ¤ x1/p−2 £ (i+2) = (−1)i uψ (u) − (p − 1)ψ (i+1) (u) p2 ¯ (i+1) ¯ ¯ ¯i x1/p−2 h ¯ψ ¯ − u¯ψ (i+2) (u)¯ , = (1 − p) (u) p2
h00 (x) = (−1)i
00 where u =¯ x1/p . When ¯ p¯ ≥(i)1, we ¯ have h (x) ≤ 0. It was proved in [42] that the (i+1) function x¯ψ (x)¯ − α¯ψ (x)¯ is completely ¯ (i) ¯ monotonic ¯ (i+1) in¯ (0, ∞) if and only if ¯ ¯ 0 ≤ α ≤ i ∈ N and that the function α ψ (x) − x¯ψ (x)¯ is completely monotonic in (0, ∞) if and only if α ≥ i + 1. A function f is called completely monotonic on an interval I if f has derivatives of all orders on I and 0 ≤ (−1)k f (k) (x) < ∞ for all k ≥ 0 on I, see [3, 25, 50]. This means that if 1 − p ≤ i + 1 for i ≥ 0 then h00 (x) ≤ 0 and that if 1 − p ≥ i + 2 for i ≥ 0 then h00 (x) ≥ 0. In conclusion, for i ≥ 0, if p ≥ −i then h00 (x) ≤ 0, if p ≤ −i − 1 then h00 (x) ≥ 0. It was obtained in [6] (see also [4, p. 274, Lemma 2]) that if g is strictly monotonic, f is strictly increasing and f ◦ g −1 is convex (or concave, respectively) on an interval I, then µ ¶ µ ¶ Z t Z t 1 1 −1 −1 g g(u) du ≤ f f (u) du (15) t−s s t−s s
holds (or reverses, respectively) for s, t ∈ I. Therefore, when p ≤ −i − 1 for i ≥ 0, inequality Z (−1)i t (i) ψ (u) du (16) (−1)i ψ (i) (Lp (s, t)) ≤ t−s s holds for positive numbers s and t; when p ≥ −i for i ≥ 0, inequality (16) reverses. The proof of Theorem 1 is complete. ¤
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Proof of Theorem 2. Taking logarithm on all sides of (11) yields Z s ln Γ(x + s) − ln Γ(x + t) 1 ψ(Lp (s, t; x)) < = ψ(x + u) du < ψ(Lq (s, t; x)) s−t s−t t which is the same as inequality (10) for the case of i = 0. The proof is complete. Proof of Theorem 3. Easy calculation gives · ¸p−1 ∂Lp (s, t; x) Lp−1 (s, t; x) = . ∂x Lp (s, t; x)
¤
(17)
Since the generalized logarithmic mean Lp (a, b) is strictly increasing in p, hence ∂Lp (s,t;x) T 1 if p S 1. It is clear that the derivative of the function defined by (12) ∂x equals · ¸ Z t ∂Lp (s, t; x) 1 − ψ (i+1) (x + u) du Qp,i,s,t (x) = (−1)i ψ (i+1) (Lp (s, t; x)) ∂x t−s s Z t ¯ (i+1) ¯ ∂Lp (s, t; x) ¯ (i+1) ¯ 1 ¯ψ = ¯ψ − (Lp (s, t; x))¯ (x + u)¯ du ∂x t−s s Z t ¯ ¯ ¯ (i+1) ¯ 1 ¯ψ T ¯ψ (i+1) (Lp (s, t; x))¯ − (x + u)¯ du t−s s if p S 1. Combining this with Theorem 1 yields that if p ≤ 1 and p ≤ −i − 2 the derivative of (12) is non-negative and that if p ≥ 1 and p ≥ −i − 1 the derivative of (12) is non-positive. Consequently, when p ≤ −i − 2 the function (12) is increasing, when p ≥ 1 the function (12) is decreasing in x > − min{s, t}. In [1, p. 260, 6.4.10], the following formula is listed: For z 6= 0, −1, −2, . . . and n ∈ N, ∞ X 1 ψ (n) (z) = (−1)n+1 n! (18) . (z + k)n+1 k=0
Further considering (17) gives ) (· ¸p−1 Z t ∞ X 1 1 1 Lp−1 (s, t; x) Qp,i,s,t (x) = (i + 1)! − du i+2 Lp (s, t; x) t − s s (x + u + k)i+2 [Lp (s, t; x) + k] k=0 (· ) ¸p−1 ∞ X Lp−1 (s, t; x) 1 1 = (i + 1)! − i+2 Lp (s, t; x) [L−(i+2) (s, t; x + k)]i+2 [Lp (s, t; x) + k] k=0 ( ¸p−1 · ¸i+2 ) · ∞ X (i + 1)! Lp (s, t; x) + k Lp−1 (s, t; x) = − . i+2 Lp (s, t; x) L−(i+2) (s, t; x + k) [Lp (s, t; x) + k] k=0 Inequality (17) implies that the function Lp (s, t; x + k) − k is increasing in k for p < 1. Thus, inequality Lp (s, t; x) ≤ Lp (s, t; x + k) − k < A(s, t; x)
(19)
holds for p < 1 and k ≥ 0. This means that Lp (s, t; x + k) Lp (s, t; x) + k ≤ L−(i+2) (s, t; x + k) L−(i+2) (s, t; x + k) and
(20)
REFINEMENTS AND GENERALIZATIONS OF KERSHAW’S DOUBLE INEQUALITY
·
Lp−1 (s, t; x) Lp (s, t; x)
¸p−1
¸i+2 Lp (s, t; x) + k − L−(i+2) (s, t; x + k) ¸p−1 · ¸i+2 · Lp−1 (s, t; x) Lp (s, t; x + k) ≥ − Lp (s, t; x) L−(i+2) (s, t; x + k) · ¸p−1 · ¸−(i+2) L−(i+2) (s, t; x + k) Lp−1 (s, t; x) = − Lp (s, t; x) Lp (s, t; x + k) · ¸p−1 · ¸−(i+2) L−(i+2) (s, t; x + k) Lp−1 (s, t; x) − ≥ Lp (s, t; x) L−(i+1) (s, t; x + k) · ¸p−1 · ¸−(i+2) L−(i+2) (s, t; x) Lp−1 (s, t; x) ≥ − Lp (s, t; x) L−(i+1) (s, t; x)
5
·
(21)
for −i − 2 < p < 1, where the following fact is used in the final line above: · ¸ · ¸ ∂ Lq (s, t; x) Lq (s, t; x) 1 1 = + > 0, ∂x Lp (s, t; x) Lp (s, t; x) E(p, p + 1; x + s, x + t) E(q, q + 1; x + s, x + t) (22) where E(p, q; a, b) is defined for p, q ∈ R and a, b > 0 by · ¸1/(q−p) p bq − aq E(p, q; a, b) = · p , pq(p − q)(a − b) 6= 0; q b − ap · ¸1/p 1 bp − ap E(p, 0; a, b) = · , p(a − b) 6= 0; p ln b − ln a · p ¸1/(ap −bp ) 1 aa , p(a − b) 6= 0; E(p, p; a, b) = 1/p bp b e √ E(0, 0; a, b) = ab, a 6= b; E(p, q; a, a) = a,
a = b.
It is remarked that the monotonicity, Schur-convexity, logarithmic convexity, comparison, generalizations, applications and history of the extended mean values E(p, q; a, b) have been investigated in many articles such as [8, 15, 16, 17, 18, 24, 26, 27, 28, 29, 31, 33, 37, 43, 44, 45, 46, 47, 48, 51, 52, 53] and the references listed in [4, pp. 393–399]. As a result, the function Q−(i+1),i,s,t (x) is non-negative, and then the function (12) for p = −(i + 1) is increasing in x. The proof of Theorem 3 is complete. ¤
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[48] H.-N. Shi, Sh.-H. Wu, and F. Qi, An alternative note on the Schur-convexity of the extended mean values, Math. Inequal. Appl. 9 (2006), no. 2, 219–224. B` udˇ engsh`ı Y¯ anji¯ u T¯ ongx` un (Communications in Studies on Inequalities) 12 (2005), no. 3, 251–257. 2 [49] J. G. Wendel, Note on the gamma function, Amer. Math. Monthly 55 (1948), no. 9, 563–564. 1 [50] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946. 2 [51] A. Witkowski, Convexity of weighted extended mean values, RGMIA Res. Rep. Coll. 7 (2004), no. 2, Art. 10; Available online at http://rgmia.vu.edu.au/v7n2.html. 2 [52] A. Witkowski, Weighted extended mean values, Colloq. Math. 100 (2004), no. 1, 111–117. RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 6; Available online at http:/rgmia.vu.edu.au/ v7n1.html. 2 [53] S.-L. Zhang, Ch.-P. Chen and F. Qi, Another proof of monotonicity for the extended mean values, Tamkang J. Math. 37 (2006), no. 3, 207–209. 2 (F. Qi) Research Institute of Mathematical Inequality Theory, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China E-mail address:
[email protected],
[email protected],
[email protected],
[email protected],
[email protected],
[email protected] URL: http://rgmia.vu.edu.au/qi.html
